Subtracting mixed numbers​

Subtracting mixed numbers (also called mixed fractions) is a key skill in basic math that combines whole number subtraction with fraction subtraction – and it’s often the trickiest part of fraction work for learners. Mixed numbers (e.g., 3 1/2, 5 3/4) include a whole number and a proper fraction, and subtracting them requires extra steps (like borrowing) when the fractional part of the first number is smaller than the second. This guide breaks down subtracting mixed numbers in plain language, with step-by-step examples for all scenarios, real-world use cases, and tips to avoid common errors.

Key Terms to Know Before Subtracting Mixed Numbers

Before diving into subtracting mixed numbers, clarify these foundational terms to avoid confusion:

  • Mixed Number: A whole number + a proper fraction (e.g., 2 3/8 = 2 + 3/8; the fraction part has a numerator < denominator).
  • Proper Fraction: A fraction where the top number (numerator) is smaller than the bottom number (denominator) (e.g., 3/4, 5/7).
  • Like Denominators: Fractions with the same denominator (e.g., 1/4 and 3/4 – easy for subtraction).
  • Unlike Denominators: Fractions with different denominators (e.g., 1/2 and 3/5 – require a common denominator first).
  • Least Common Denominator (LCD): The smallest number both denominators divide into evenly (e.g., LCD of 2 and 5 is 10).
  • Borrowing (Regrouping): Converting 1 whole number into a fraction to subtract a larger fractional part (e.g., 3 1/4 → 2 5/4 when borrowing 1).

How to Subtract Mixed Numbers (Two Reliable Methods)

There are two main ways to subtract mixed numbers – choose the method that works best for your skill level:

Method 1: Subtract Whole Numbers and Fractions Separately (Most Intuitive)

This method is ideal for subtracting mixed numbers with like denominators, or unlike denominators after finding the LCD.

Step-by-Step for Method 1

  1. Find the LCD (if denominators are unlike): Rewrite the fractional parts with the same denominator.
  2. Check if borrowing is needed: If the first fraction < the second fraction (e.g., 3 1/4 – 1 3/4), borrow 1 from the whole number of the first mixed number.
  3. Subtract the whole numbers: Subtract the second whole number from the first (adjusted for borrowing if needed).
  4. Subtract the fractions: Subtract the second fraction from the first (keep the LCD).
  5. Combine the results: Simplify the fraction (if needed) and combine with the whole number.

Example 1: Subtracting Mixed Numbers (Like Denominators, No Borrowing)

Problem: 5 3/8 – 2 1/8Solution:

  1. Denominators are like (8) – no LCD needed.
  2. 3/8 > 1/8 – no borrowing needed.
  3. Subtract whole numbers: 5 – 2 = 3.
  4. Subtract fractions: 3/8 – 1/8 = 2/8 = 1/4.
  5. Combine: 3 1/4.

Example 2: Subtracting Mixed Numbers (Like Denominators, With Borrowing)

Problem: 4 1/5 – 1 3/5Solution:

  1. Denominators are like (5) – no LCD needed.
  2. 1/5 < 3/5 – borrow 1 from 4 (4 → 3, 1/5 → 6/5).
  3. Subtract whole numbers: 3 – 1 = 2.
  4. Subtract fractions: 6/5 – 3/5 = 3/5.
  5. Combine: 2 3/5.

Example 3: Subtracting Mixed Numbers (Unlike Denominators, With Borrowing)

Problem: 6 1/2 – 2 3/4Solution:

  1. LCD of 2 and 4 is 4 – rewrite fractions: 1/2 = 2/4, so 6 1/2 = 6 2/4.
  2. 2/4 < 3/4 – borrow 1 from 6 (6 → 5, 2/4 → 6/4).
  3. Subtract whole numbers: 5 – 2 = 3.
  4. Subtract fractions: 6/4 – 3/4 = 3/4.
  5. Combine: 3 3/4.

Method 2: Convert to Improper Fractions (Most Reliable for Complex Problems)

This method eliminates the need for borrowing by converting mixed numbers to improper fractions first – perfect for tricky problems.

Step-by-Step for Method 2

  1. Convert both mixed numbers to improper fractions: Formula = (Whole number × denominator) + numerator (e.g., 3 1/2 = (3×2)+1 / 2 = 7/2).
  2. Find the LCD (if needed): Rewrite the improper fractions with the same denominator.
  3. Subtract the fractions: Subtract the second numerator from the first (keep the LCD).
  4. Convert back to a mixed number: Divide the numerator by the denominator (remainder = new numerator).
  5. Simplify (if needed): Reduce the fraction to lowest terms.

Example: Subtracting Mixed Numbers (Improper Fraction Method)

Problem: 7 2/3 – 4 5/6Solution:

  1. Convert to improper fractions:
    • 7 2/3 = (7×3)+2 / 3 = 23/3
    • 4 5/6 = (4×6)+5 / 6 = 29/6
  2. LCD of 3 and 6 is 6 – rewrite 23/3 = 46/6.
  3. Subtract fractions: 46/6 – 29/6 = 17/6.
  4. Convert to mixed number: 17 ÷ 6 = 2 with remainder 5 → 2 5/6.
  5. Simplify: 5/6 is already reduced – final result = 2 5/6.

Real-World Examples of Subtracting Mixed Numbers

Subtracting mixed numbers isn’t just for math class – it’s critical for everyday tasks that involve measurements:

Example 1: Cooking/Baking

Problem: A recipe needs 4 1/3 cups of flour. You only have 2 2/3 cups. How much more flour do you need?Solution: 4 1/3 – 2 2/3 = 3 4/3 – 2 2/3 = 1 2/3 cups.

Example 2: Measuring Length (Woodworking/DIY)

Problem: A board is 8 1/4 feet long. You cut off 3 3/8 feet. How long is the remaining board?Solution:

  1. LCD of 4 and 8 is 8 → 8 1/4 = 8 2/8.
  2. Borrow 1 from 8 → 7 10/8.
  3. Subtract: 7 10/8 – 3 3/8 = 4 7/8 feet.

Example 3: Time Management

Problem: You study for 3 3/4 hours on Saturday and 1 5/6 hours on Sunday. How much longer did you study on Saturday?Solution:

  1. LCD of 4 and 6 is 12 → 3 3/4 = 3 9/12, 1 5/6 = 1 10/12.
  2. Borrow 1 from 3 → 2 21/12.
  3. Subtract: 2 21/12 – 1 10/12 = 1 11/12 hours.

Example 4: Weight (Cooking/Shipping)

Problem: A package weighs 5 2/5 pounds. After removing a 1 3/4 pound item, what’s the new weight?Solution:

  1. LCD of 5 and 4 is 20 → 5 2/5 = 5 8/20, 1 3/4 = 1 15/20.
  2. Borrow 1 from 5 → 4 28/20.
  3. Subtract: 4 28/20 – 1 15/20 = 3 13/20 pounds.

Common Mistakes to Avoid When Subtracting Mixed Numbers

These errors are the most common when subtracting mixed numbers – watch for them!

  1. Forgetting to Find the LCD: Subtracting fractions with unlike denominators directly (e.g., 1/2 – 1/3 = 0/-1) leads to wrong answers.
  2. Borrowing Incorrectly: When borrowing 1 whole number, convert it to a fraction with the same denominator (e.g., 1 = 4/4, not 1/4).
  3. Skipping Simplification: Leaving fractions like 4/8 instead of simplifying to 1/2 makes answers incomplete.
  4. Subtracting Whole Numbers Before Checking Fractions: Always check if the fraction needs borrowing before subtracting whole numbers (e.g., 3 1/4 – 1 3/4 ≠ 2 – 2/4).
  5. Miscalculating Improper Fractions: When converting mixed numbers, use (Whole × Denominator) + Numerator (e.g., 2 1/3 = 7/3, not 5/3).

Frequently Asked Questions (FAQs) About Subtracting Mixed Numbers

Q1: Can I subtract a mixed number from a whole number (e.g., 6 – 2 1/4)?

A1: Yes – convert the whole number to a mixed number: 6 = 5 4/4. Then subtract: 5 4/4 – 2 1/4 = 3 3/4.

Q2: What if the result is an improper fraction after subtraction?

A2: Convert it to a mixed number (e.g., 11/4 = 2 3/4) – mixed numbers are the standard form for answers.

Q3: Do I have to use the LCD, or can I use any common denominator?

A3: You can use any common denominator (e.g., 24 instead of 12 for 4 and 6), but the LCD makes calculations simpler.

Q4: How do I subtract mixed numbers with negative numbers (e.g., -3 1/2 – 1 3/4)?

A4: Convert to improper fractions: -7/2 – 7/4 = -14/4 – 7/4 = -21/4 = -5 1/4.

Q5: What’s the easiest way to remember borrowing for mixed numbers?

A5: Think of 1 whole as “denominator over denominator” (e.g., 1 = 5/5, 1 = 8/8) – add that to the fraction part before subtracting.

Conclusion

Subtracting mixed numbers is a skill that becomes second nature with practice – the key is to either subtract whole numbers/fractions separately (with borrowing if needed) or convert to improper fractions for complex problems. Whether you’re cooking, measuring wood, or studying for a test, knowing how to subtract mixed numbers is a practical tool that applies to everyday life. Remember to check for common denominators, borrow when the fraction is too small, and simplify your answer – and you’ll master this skill in no time.

If you have questions about subtracting specific mixed numbers, or need help with a tricky borrowing problem, leave a comment below!